While the unconstrained version of this problem – i.e., Max-AND with no negations – is trivially polynomial time solvable, the additional simple cardinality constraint makes the problem significantly harder. In particular, Raghavendra and Steurer [1]} showed that assuming the Small Set Expansion Hypothesis (SSEH), \({\sf D}k{\sf S}\)  is -hard to approximate to any constant factor. Furthermore, Manurangsi [2]} showed assuming the Exponential Time Hypothesis (ETH), there are no polynomial time algorithms which can approximate \({\sf D}k{\sf S}\)  up to an almost polynomial ratio. A similar phenomena was also observed by Austrin and Stankovic [3]} for the setting of cardinality constrained Max-Cut as well. Furthermore, the nature of how the approximability of the CSP is affected is also predicate dependent. For instance, while in the case of D\(k\) S, the cardinality constraint makes it constant factor inapproximable for any constant, in the case of Max-Cut, there exists a \(0.858\) -approximation factor for any \(k = \mu n\)  with \(\mu \in (0,1)\) . Hence we are motivated to ask the following question:
